9. (a) Evaluate fFdR where F = 4zi - 2rj+2rk, and C is the intersection of the cylinder 1² + y² = 1 and the plane 2y+ 1 (oriented counter clockwise as seen by a person standing on the plane). (b) Verify (a) by using Stokes' Theorem.

(2) Write parametrizations 7(u, v) for the following surfaces (a) a sphere with radius 5 (b) a cone (c) a cylinder (3) Write the tangent vectors r, and r, for the parameterizations in the last problem.

A) Find an equation of a line tangent the circle of radius 3 with its center P(1, 3, 4) located in a plane parallel to the xy-plane at t = 2n. 2. B) Calculate velocity and acceleration for a particle travelling on the circle of radius 3 with its center P(1, 3, 4) located in a plane parallel to the xy-plane at t = 2n. C) The Projection of u along v is - 3. Calculate the projection of acceleration along velocity at t = 2n.

A particle moves in space along a path whose parametric equations are given by x1 = bsin(wt), x2 = bcos(wt), x3 = c, where b and c are constants. a) Find its position vector r, velocity v, and acceleration a at any time t. b) Show that the acceleration is perpendicular to the velocity and the x3 - axis.

Find the distance from the point (1,1,1) to the plane 2x+2y+z=0

What is the perpendicular vector to the surface f(x.y.a)-ay-3z+y-0 at (1,-12)? (3,3.0) Ob. (1,1.3) (-1,-1-3) (4-10)

4. Consider the vector function r(z, y) (r, y, r2 +2y"). (a) Re-write this vector function as surface function in the form f(1,y). (b) Describe and draw the shape of the surface function using contour lines and algebraic analysis as needed. Explain the contour shapes in all three orthogonal directions and explain and label all intercepts as needed. (c) Consider the contour of the surface function on the plane z= for this contour in vector form. 0. Write the general equation